P-Divisible Groups Introduction

p-divisible groups

Michael Lipnowski

Introduction

We’d like to understand abelian schemes A over an arbitrary base S, or at least over rings of integers and localizations and completions thereof. That’s very difficult. As a first step, we pv consider its associated torsion group schemes. Let Gv = ker(A −→ A); we’ll justify later that these are commutative finite flat groups schemes over S. They fit in an exact sequence

v iv p 0 → Gv −→ Gv+1 −→ Gv+1,

iv where Gv −→ Gv+1 is the natural inclusion. Furthermore, the order of Gv, as a group scheme over S, is p2gv, where g is the relative dimension of A over S. These are the characterizing properties of a p-divisible groups over S of height 2g. This write up is devoted to some basic but important aspects of the structure theory of p-divisible groups. The most exciting thing discused will be the relation between connected p-divisible groups and divisible formal Lie groups over R.

Connection with Faltings’ Proof This section is very rough and intended for motivation only. Please take it with a grain of salt, since I don’t properly understand the contents myself. Large parts might even be wrong.

The main goals of our seminar are proving (for a number ﬁeld K):

• Weak Tate Conjecture. The map

t {Abelian varieties of dimension g/K} −→{Galois representations GK → GL2g(Qp)}

has ﬁbers consisting of isogeny classes.

• Isogeny classes are ﬁnite.

The key will be a careful analysis of Faltings height h,ˆ namely that “Faltings height doesn’t change much under isogeny”. Though we haven’t deﬁned Faltings height yet, it will be roughly −1 volume(A(K ⊗Q C)) , where this volume is normalized by a basis of Neron diﬀerentials (an integral structure on the 1-forms of A). It will turn out that this height is controlled by discriminants of various p-divisible groups of A.

Consider an abelian varieties A/K of fixed dimension g with good reduction outside a fixed finite set Σ of places of K. By semi-stable reduction, there is a fixed finite field extension K0/K, 0 depending only on g and Σ so that A ⊗K K has semi-stable reduction at places of bad reduction (a theorem to be discussed in the winter).

1 The change in height under isogenies f : A → A0 between *semistable* abelian varieties is therefore the main focus. Assume A has good reduction at v, an l-adic place of K, and f is an l-isogeny. Raynaud’s computation of the Galois action on det(ker(f))/K0 for l-torsion groups over l-adic integer rings gives an explicit expression for the change in Faltings height, when e(v) ≤ l − 1. So while Raynaud’s calculation of the discriminant applies for large l, relative to the absolute ramiﬁcation of K0/Q, it won’t suﬃce for small l, in particular for small l of bad reduction. Tate’s computation of the discriminant of constitutents of a p-divisible groups over a complete, local, noetherian ring will be crucial to bypass this problem.

Deﬁnitions

iv A p-divisible group of height h over a scheme S is an inductive system G = ({Gv},Gv −→ Gv+1)v≥1 of group schemes over S satisfying the following:

hv (i) Gv is a ﬁnite locally free commutative group scheme over S of order p .

v iv p (ii) 0 → Gv −→ Gv+1 −→ Gv+1 is an exact sequence of group schemes.

For ordinary abelian groups, these axioms imply that ∼ v h h Gv = ( /p ) and lim Gv = ( p/ p) . Z Z → Q Z 0 A homomorphism of p-divisible groups f : G = (Gv, iv) → H = (Hv, iv) is a system of homomorphisms of group schemes fv : Gv → Hv which are compatible with the structure maps: 0 ivfv = fv+1iv. Let iv,m : Gv → Gm+v denote the closed immersion iv+m−1 ◦ ... ◦ iv+1 ◦ iv. pm A diagram chase shows that Gm+v −→ Gm+v can be factored uniquely through iv,m via a map m jm,v : Gm+v → Gv (so im,v ◦ jm,v = p ) and furthermore that

iv,m jm,v 0 → Gm −−→ Gm+v −−→ Gv is exact. But since orders are multiplicative in exact sequences, a consideration of orders shows that

iv,m jm,v 0 → Gm −−→ Gm+v −−→ Gv → 0 is exact (i.e. the closed immersion coker(iv,m) → Gv is an isomorphism). Here are some examples of p-divisible groups.

(a) The simplest p-divisible group over S of height h is the constant group:

h v h iv v+1 h (Qp/Zp) = ((Z/p ) −→ (Z/p ) )

with iv the natural inclusion (using multiplication by p).

(b) The p-divisible group of the multiplicative group Gm/S is

v iv v+1 Gm(p) = (Gm[p ] −→ Gm[p ])

v v where Gm[p ] is the p -torsion subgroup scheme of Gm and the iv are the natural inclusions. To be precise, the inclusions

2 v pv pv+1 v+1 Gm[p ](T ) = {f ∈ O(T ): f = 1} → {f ∈ O(T ): f = 1} = Gm[p ](T )

v v+1 are functorial in S-schemes T and so deﬁnes a map iv : Gm[p ] → Gm[p ].

v pv – Over an affine SpecR ⊂ S, Gm[p ]/R is just SpecR[x]/(x − 1), which is free over R of v v v rank p . Hence, Gm[p ] is finite flat over S of order p . –

v v+1 p v+1 pv+1 pv ker(Gm[p ] −→ Gm[p ])(T ) = {f ∈ O(T ): f = 1 with f = 1} v+1 iv(T ) v+1 = image{Gm[p ](T ) −−−→ Gm[p ]}

v v+1 p v+1 functorially in S-schemes T. Thus, iv is the kernel of Gm[p ] −→ Gm[p ].

It follows that Gm(p) is a p-divisible group of height 1 over S. (c) What follows is the most important example, alluded to in the introduction. Let A/S be an abelian scheme of relative dimension g. That is, A is a proper, smooth group scheme over S whose geometric ﬁbers are connected of dimension g. The hypotheses imply that A has commutative multiplication. Let [n]: A → A denote multiplication by n ∈ Z − {0}.

– For any geometric point s : SpecΩ → A, the fiber [n]s : As → As is a finite map. Thus, [n] is quasifinite and proper, and thus is finite. In particular, A[n]/S is finite.

– The structure map A → S is flat and for any geometric point s, [n]s : As → As is flat. Thus, by the fibral flatness theorem, [n] is flat. In particular, A[n]/S is flat.

Thus, A[n]/S is a finite flat group scheme. It is also finitely presented since A/S is, and so it is locally free over S. Because all of its geometric fibers have order n2g, by theory for abelian varieties over an algebraically closed field, A[n]/S has order n2g. v v+1 Also, as explained for Gm, the natural inclusion iv : A[p ] → A[p ] is the kernel of pv A[pv+1] −→ A[pv+1]. v Thus, A(p) = (A[p ], iv) is a p-divisible group of height 2g over S.

Etale and Connected Groups

In topology, we know that for reasonable connected spaces S and a choice of base point α ∈ S, there is a functorial bijection between coverings of S and actions of π1(S, α) on ﬁnite sets realized through the deck transformation action on the ﬁber over α. Remarkably, the same correspondence often carries over algebraically.

Let S be connected and locally noetherian with geometric point α : SpecΩ → S. Let Fet/S denote the category of ﬁnite etale coverings of S.

There is a functor

3 D : Fet/S → Sets Y 7→ Y (α) where Y (α) denotes the set of geometric points of a finite etale covering Y → S lying over α. This functor has an automorphism group π = π1(S, α), the fundamental group of S with basepoint α. It is naturally a profinite group. Let Fπ-Sets denote the category of finite sets with a continuous action of π. By construction, each Y (α) carries an action of π. Thus, we can view D as a functor

D : Fet/S → Fπ-Sets. D has two good properties: it commutes with products and disjoint unions. This is especially useful in light of Theorem (Grothendieck). D : Fet/S → Fπ-Sets is an equivalence of categories. We can describe the inverse equivalence in certain cases of interest. • If S = Spec(k) for a ﬁeld k, then a geometric point is an embedding α : k → Ω. Autk(Ω) acts on Y (α) = HomS(SpecΩ,Y ). This action gives a map Autk(Ω) → π which factors through Gal(ks/k). The induced map Gal(ks/k) → π is an isomorphism. The reverse equivalence is given by

s X 7→ Spec(Mapsπ(X, k )).

• If S = SpecR is local henselian with closed point s, then Y 7→ Ys is an equivalence of categories

FEt/S → FEt/s.

Thus, we have the exact same description of the inverse equivalence in this case:

sh X 7→ Spec(Mapsπ(X,R )),

where Rsh is the strict henselization of R which is compatible with our choice of ks/k. We can restrict this equivalence to the subcategory FGet/S of finite etale group schemes with maps of groups between them. Let G/S be a group scheme with m, e, i denoting the multiplication, inversion, and identity morphisms respectively. G(α) is a group whose underlying set carries an action of π. But we say much more! G ×S G is also finite etale over S. Thus, π acts through the automorphism group of G(α). We denote Fπ-Gps the category of finite groups with a continuous action of π.

Also, if we let D0 denote the inverse equivalence to D, then H = D0(G(α)) is a ﬁnite etale group scheme over S. Indeed, because D commutes with products and is full, there is a morphism m lifting the multiplication of G(α). Similarly, there are morphisms e and i lifting inversion and the identity. They satisfy the appropriate commutativity because D is faithful. It follows that by restricting D to FGet/S gives an equivalence of categories

4 D : FGet/S → Fπ-Gps. This equivalence gives us a fairly completely understanding of finite etale S-groups. As a toy example, we can use this equivalence to prove that the only connected etale group is trivial. Because D is an equivalence of categories which commutes with disjoint unions, connected finite etale S-schemes Y are exactly those for which π acts transitively on Y (α). But as explained above, for G a finite etale group scheme, π acts on G(α) as group automorphisms. In particular, it preserves the identity of G(α). Thus, π permutes G(α) transitively iff G(α) is a singleton, which implies that G is trivial.

Now let S = SpecR, where R is a Henselian local ring. That way, the connected component G0 of the identity section lifts the identity component of the special fiber, so its formation commutes with products (and local henselian base change) and hence it is a finite flat group scheme over S. For any finite flat group scheme G, define Get = G/G0. Because its identity section is an open immersion, it is an etale group scheme. Further, if f : G → H is any map with H etale, then G0 factors through H0, which is trivial by our above discussion. Thus, we have the exact sequence

j 0 → G0 −→i G −→ Get → 0. j with G0 −→i G, G −→ Get characterized by the following universal properties: • If f : H → G is any map from a connected finite flat group scheme over S, then f factors uniquely through i. • If f : G → H is any map to a finite etale group scheme over S, then f factors uniquely through j. Thus, G 7→ G0,G 7→ Get are actually a functors on finite flat commutative S-group schemes. Over a field, we can prove that these functors are exact as follows. j Suppose 0 → K −→i G −→ H → 0 is an exact sequence of finite flat commutative groups over R. Then we have the following short exact triple of complexes:

0 j0 0 −−−→ K0 −−−→i G0 −−−→ H0 −−−→ 0 (A•)       y y y j 0 −−−→ K −−−→i G −−−→ H −−−→ 0 (B•)       y y y et jet 0 −−−→ Ket −−−→i Get −−−→ Het −−−→ 0 (C•) We readily check that i0 is the kernel of j0 and that jet is the cokernel of iet.

j Let α be a closed geometric point of S. Note that since K −→i G is the functorial kernel of G −→ H, we have that 0 → K(α) → G(α) → H(α) is exact. But A(α) = Aet(α) for any finite flat et commutative group scheme A. Thus, by the equivalence of categories, Ket −→i Get is the kernel of jet Get −→ Het in the category of etale groups over S. But any map from a finite group A to the etale group Get which is killed by jet must factor uniquely through Aet and so uniquely through Ket. It follows that the bottom sequence is exact. By a comparison of orders, the top row must be left exact too.

5 f • In general, any left exact sequence 0 → A −→i B −→ C is a left exact sequence of finite flat commutative group schemes over R with order(B) = order(A)order(C) is exact. q The induced map f : B/A → C has kernel 0. Indeed, B −→ B/A is faithfully flat, so if x ∈ ker(f)(T ), thenx, ˜ the base change of x via some faithfully flat T 0 → T, has a preimage y ∈ B(T 0). But then y ∈ ker(f)(T 0) = A(T 0), whencex ˜ = 0 since the composition q A −→i B −→ B/A is zero. This implies that x = 0. Hence, the homomorphism f is a closed immersion between two finite flat R-groups of the same order, and so is an isomorphism. Thus, f = f ◦ q is faithfully flat.

In conclusion, we see that G 7→ G0,G 7→ Get are exact functors.

Now let G = (Gv, iv) be a p-divisible group. By the exactness proved above, 0 0 0 et et et G := (Gv, iv),G := (Gv , iv ) are p-divisible groups. The connected etale sequence at ﬁnite level

0 et 0 → Gv → Gv → Gv → 0 leads to the exact sequence of p-divisible groups

0 → G0 → G → Get → 0

For ﬁnite groups G over a perfect ﬁeld k of characteristic p, the connected etale sequence

0 → G0 → G → Get → 0. is particularly nice. The map G → Get admits a section. Indeed, over such k, Gred is etale over k. Thus, Gred × Gred is reduced, and so multiplication Gred × Gred → G factors through Gred, whence Gred is a closed subgroup scheme of G. Since Gred(α) = G(α) = Get(α) for a geometric point α, we conclude Gred = Get. The closed immersion Gred → G is a section. For ﬁnite groups over a general henselian local base, G → Get does not admit a section. But this much is true: Call a p-divisible group G ordinary if the ﬁnite level terms of G0 are multiplicative, i.e. have etale Cartier duals.

Theorem. Let G be a p-divisible group over a perfect ﬁeld k of characteristic p. Let R be any henselian local ring with residue ﬁeld k. Then G admits a unique lift G˜ to R with split connected-etale sequence (necessarily lifting the connected-etale sequence of G).

This plays an important role in the deformation theory of abelian varieties.

Etale p-divisible groups At this point, it seems worthwhile to mention that this equivalence of categories extends to etale p-divisible groups over a henselian local ring R with closed geometric point α. Suppose we have an etale p-divisible group G = (Gv, iv) of height h over R. Each Gv(α) is isomorphic to Z/pvZ. We can form the inverse limit over multiplication by p maps

G(α) := lim Gv(a). ←

6 The multiplication by p maps are compatible with the action of π1(R, α). Thus, G(α) still carries ∼ v h a continuous action of π1(R, α). Furthermore, since Gv(α) = (Z/p Z) , it follows that G(α) is a free Zp module of rank h.

Furthermore, suppose M is a free Zp module of rank h with a continuous action of π. Then the quotient M/pvM is a group of order pvh with a continuous action of π on it. Thus, by the equivalence of categories, there is an etale group over R, say Gv, corresponding to it. The Galois v v v+1 p p compatible multiplication map M/p M −→ M/pM, which is induced by Gv+1 −→ G1, has kernel M/pvM −→iv M/pv+1M. By the equivalence of categories, there is a corresponding map iv Gv −→ Gv+1. Thus, G = (Gv, iv) is an etale p-divisible group. As a consequence of the above, we conclude

Theorem. Let pdivEt/R be the category of etale p-divisible groups of over R, and let Free-π Mod/Zp denote the category of free Zp modules with a continuous action of π. The functor G 7→ G(α) is an equivalence of categories

pdivEt/R → Free-π Mod/Zp(h). carrying height to Zp rank. As usual, the most important instance of this is for abelian schemes.

• Let A/R be an abelian scheme. Then if l 6= p, A(l) is an etale l-divisible group. The corresponding Zl module is none other than Tl(A) with action of π1(R, α).

Then enter Grothendieck, who proved an amazing result.

Theorem (Grothendieck). Let R be a henselian dvr with fraction ﬁeld K, and let l be a rational prime. An abelian variety A/K extends to an abelian scheme /R iﬀ A(l) extends to an l-divisible group over R.

This is proven through the theory of Neron models and the semi-stable reduction theorem. The deepest case is when l is the residue characteristic.

As an example of the power of this result, note that we can recover the classical Neron-Ogg-Shafarevic criterion.

Theorem. Let p be the residue characteristic of R. Choose any l 6= p be a prime, A an abelian variety over K. Then A has good reduction at l iﬀ the Galois representation on Tl(A) is unramiﬁed at l.

Proof. If A(l) admits a prolongation to R, then it is etale since each A[ln] has order a power of l, which is prime to the residue characteristic. But by the equivalence of categories for etale l-divisible groups, A(l) admits a prolongation iff the action of π1(K, α) on the corresponding Zl-module, which is none other than Tl(A), factors through π1(R, α), i.e. iff the Galois action on Tl(A) is unramified.

7 Serre-Tate Equivalence for Connected p-divisible groups and Divisible Formal Groups

Let R be a complete, noetherian, local ring with residue field k of characteristic p > 0. And let A = R[[X1, ..., Xn]] be the power series ring over R in n variables. Definition. An n dimensional commutative formal Lie group Γ over R is a pair of maps m : A → A⊗ˆ RA (the completed tensor product being given the max-adic topology), e : A → R such that m is coassociative and cocommutative with e as a counit. Concretely, e is given by Xi 7→ 0 and m is given by a family f(Y,Z) = (fi(Y,Z)) of n power series in 2n variables such that (i) X = f(X, 0) = f(0,X). (ii) f(X, f(Y,Z)) = f(f(X,Y ),Z). (iii) f(X,Y ) = f(Y,X). There is a much more functorial interpretation of formal Lie groups over R which is convenient for many purposes. Namely, a formal group over R is a group object in the category of formal schemes over R and a formal Lie group over R is formally smooth formal group over R. In other words, a formal group is a “pro-representable” functor F : Profinite Artinian/R → Gps, and a a formal Lie group is one for which F (B) → F (B/I) is surjective for any finite artinian R-algebra B and any square zero ideal I ⊂ B. It’s a non-trivial fact that the latter condition implies that the pro-representing object is a power series ring over R. Some interesting examples: • Let G be an algebraic group over a field k with identity e. Let T be a finite artinian k-scheme. Call x ∈ G(T ) small if x is supported at e. The small T -points form a subgroup of G(T ) because e · e = e, e−1 = e. Thus, restricting G to small points determines a functor

Gb : Proﬁnite Artinian/k → Gps.

Also, any small point factors through an artinian quotient of Spec(OG,e). Thus, Gb is pro-represented by Spfk(OdG,e). In the case where G is smooth over k, one can see that OdG,e is a power series ring in n = dim G-variables and so is formally smooth, i.e. Gb is a formal Lie group. • We show how to recover the formal group law, as originally deﬁned, from the functor on small artinian points. Take G = Gm, for concreteness. n n We simply take the canonical inclusion in : Spec(OG,e/me ) ×k Spec(OG,e/me ) ⊂ G ×k G, compose with the two projection maps, and multiply the two of them together using the functor G.b Knowing abstractly that the formal group multiplication is given by a power series, the result of the above computation of proj1 ◦ in ∗ proj2 ◦ in, ∗ denoting multiplication in G(G ×k G), tells us what the coeﬃcients are up to order n. In the case of −1 Gm = Spec k[Z,Z ], reparameterizing with Z = 1 + T, (so that X 7→ 0 is the counit) we get

f(X,Y ) = X + Y + XY + ≥ nth powers. Similarly, we can recover the inversion.

8 Back to the original set up, as in the first defintion of formal Lie groups. Let X ∗ Y := f(X,Y ) = X + Y + higher powers. Define [p]Γ(X) = X ∗ ... ∗ X (p times), the homomorphism corresponding to multiplication by p in Γ. We call Γ divisible if [p]∗ : A → A is finite free.

Let I = (X1, ..., Xn) be the augmentation ideal of A. v ∗ Consider Av = A/[p ]Γ(I). This is a finite flat R module. Thus, since A satisfies the Hopf algebra axioms “at finite level”, each Γpv = Spec(Av) is a group scheme with multiplication induced by m. We claim that any finite (locally) free map φ : A → A has rank = rankRA/φ(I).

• Indeed, let a1, ..., an be a basis for A over itself via φ. We can even assume that a1 = 1. Let ai = ai mod φ(I).

– If we can ﬁnd a relation,

r1a1 + ... + rnan = 0 mod φ(I)

for some ri ∈ A, then

φ(r1)a1 + ... + φ(rn)an = φ(i) =⇒ φ(r1 − i)a1 + ... + φ(rn)an = 0 for some i ∈ I (remember that φ is an R-algebra map). This implies that r1 = i ∈ I ∩ R = {0}, r2 = ... = rn = 0 because the ai form a basis for Aφ. Thus, the ai are linearly independent over R. – By assumption, we can express any a ∈ A as

a = φ(b1)a1 + ... + φ(bn)an

for some bi ∈ A. Write each b∗ = r∗ + i∗ for r∗ ∈ R, i∗ ∈ I. Then

a = r1a1 + ... + rnan.

Thus, the ai also generate A/φ(I) as an R-module.

Our claim is thus true: φ has rank = rankRA/φ(I).

In particular, since “free over itself of degree ? ” is multiplicative in ? with respect to composition of maps,

v v rankRA/[p ](I) = (rankRA/[p](I)) . v But each Av = A/ψ (I) is a local ring. Thus, each Γpv is connected and so has order a power of p. h hv Let p be the order of Γp. Then by the above, Γpv has order p . v v But Γpv represents Γ[p ], the p -torsion functor of Γ. Thus, the canonical inclusion v v+1 iv :Γpv → Γpv+1 , realized through Yoneda via the functorial inclusion Γ[p ] ⊂ Γ[p ] is the kernel [pv] of Γpv+1 −−→ Γpv+1 .

Thus, Γ(p) = (Γpv , iv) is a connected p-divisible group. Thought of in terms of the functor of small artinian points, it is clear that the association Γ 7→ Γ(p) is functorial. The following remarkable fact is true:

9 Theorem. Let R be a complete, local, noetherian ring with maximal ideal m and residue ﬁeld k = R/m of characteristic p > 0. Then Γ 7→ Γ(p) is an equivalence between the category of divisible commutative formal Lie groups and the category of connected p-divisible groups. Proof. (Tate) Step 1: Full Faithfulness Let Γ be a divisible formal Lie group over R. It’s coordinate ring A = R[[X1, ..., Xn]] has maximal ideal M = mA + I, where I = (X1, ..., Xn). As above, we let [p]: A → A correspond to multiplication by p in Γ. First, note that the ideals mvA + [pv](I) form a neighbourhood base of 0 in A :

v v v v v •A/(m A + [p ](I)) = Av/m Av is artinian. Thus, each m A + [p ](I) is open.

2 • [p](Xi) = pXi + higher order. Thus, [p](I) ⊂ pI + I ⊂ (mA + I)I = MI. So

[pv](I) ⊂ M vI,

and these neighbourhoods are arbitrarily small.

Thus,

A = lim A/[pv](I) ← and so the functor is fully faithful.

Step 2: Reduction to R = k Let G = (Gv = Spec(Av), iv) be our connected p-divisible group with base change G = (Gv = Spec(Av)), iv) to k. Let A = lim Av, A = lim Av. ← ← Choose an augmented topological isomorphism A ' k[[X1, ..., Xn]]. Pick liftings R[[X1, ..., Xn]] → Av of the quotient maps k[[X1, ..., Xn]] Av, and arrange these choices to be compatible with change in v. This can be arranged because Av+1 → Av is a surjection between finite free R-modules. By Nakayama’s Lemma, the mapping R[[X1, ..., Xn]] → Av to a finite free R-module target is surjective, whence due to R-module splittings of the surjections Av+1 Av we get two conclusions: (i) lim Av is topologically isomorphic as an R-module to a product of countably many copies of R, ← f and (ii) the natural map R[[X1, ..., Xn]] −→ lim Av is surjective and splits in the sense of ← R-modules. By the construction of the countable product decomposition in (i) for the inverse limit as an R-module, it follows that the formation of the module splitting in (ii) is compatible with passage to the quotient modulo m. By the result assumed over the residue field, ker(f) ⊗R k = 0, i.e. m ker(f) = ker(f). Since R[[X1, ..., Xn]] is a noetherian, ker(f) is a finitely generated R[[X1, ..., Xn]]-module with M ker(f) = (mR[[X1, ..., Xn]] + I) ker(f) = ker(f). Thus by Nakayama, ker(f) = 0. Hence, our map of R-algebras R[[X1, ..., Xn]] → lim Av ← is an isomorphism. This is even a topological isomorphism: its formation commutes with passage N to quotients modulo any m (N ≥ 1), and for artinian R the ideals av = ker(R[[X1, ..., Xn]] Av)

10 are a system of open ideals (as each Av has finite length), so a beautiful theorem of Chevalley [Matsumura, Exercise 8.7] (see hint in the back of the book!) gives the cofinality of the av’s. This is the desired topological aspect for the isomorphism. As a consequence of the topological nature of the isomorphism, artinian points can be “read off” from the inverse limit description, and so we get a functorial formal group law on R[[X1, ..., Xn]] (via the group laws on the Gv’s), with the formation of this group law commuting with passage to the residue field. Hence, [p]∗ is finite flat (as this is assumed known after reduction over the residue field, and can be pulled up by m-adic completeness and the local flatness theorem). We v likewise see that Gv is the p -torsion on our formal group law over R because this is true on the level of artinian points (by construction of the formal group law on the inverse limit). Consequently, taking v = 1 shows that the finite flat map [p]∗ has degree ph, completing the argument over R (granting the results over the residue field). Thus, it suffices to prove the result for p-divisible groups over k, which is characteristic p.

Step 3: Proof when R = k The key is to use Frobenius F and Verschiebung V. Consider the category of fppf sheaves of abelian groups over k, which contains finite groups and p-divisible groups as full subcategories. p-divisible groups can be characterized as those sheaves G for which G(B) is p-power torsion for all k-algebras B, [p] is surjective, and each G[pv] is representable by a finite commutative groups schemes over k. The the functor G 7→ G(p) makes sense for finite groups, but also for p-divisible groups. This is because:

• ∗(p) is exact.

• It preserves the orders of objects represented by ﬁnite ﬂat commutative group schemes objects.

(p) (p) (p) In particular, for any p-divisible group G, G = (Gv , iv ) is a p-divisible group. We also get maps of p-divisible groups F : G → G(p) and V : G(p) → G deﬁned by the usual Frob and Ver

maps at ﬁnite level. These satisfy VF = [p]G,FV = [p]G(p) . Since [p]: G → G is a surjective map of fppf sheaves of abelian groups, if G has height h, then so does G(p) and F and V are both surjective with ﬁnite kernel of order ≤ ph.

v F (pv) Let Hv = ker(G −→ G ) = Spec(Bv).Hv ⊂ Gv and Gv ⊂ HN for some large N, since any ﬁnite connected group is killed by a suﬃciently high power of Frobenius. Thus,

A = lim Av = lim Bv. ← ←

Let Iv be the maximal ideal of Bv. Then I = lim Iv is the maximal ideal of A. ← 2 Suppose that x1, ..., xn are elements of I whose images form a k basis for I1/I1 . These elements 2 also form a k basis for Iv/Iv .

• Indeed, H1 ⊂ Hv is the kernel of F on Hv. Thus, Iv → I1 is a surjective k map with kernel (p) Iv , the ideal generated by pth powers of elements of Iv. Thus,

2 ∼ (p) 2 ∼ 2 Iv/Iv = Iv/(Iv + Iv ) = I1/I1 .

11 Consider the maps

uv : k[X1, ..., Xn] → Bv; Xi 7→ xi. pv pv v By the above, these maps are surjective. The kernel contains (X1 , ..., Xn ) because F kills Hv. Thus, the homomorphism

pv pv uv : k[X1, ..., Xn]/(X1 , ..., Xn ) → Bv is surjective. But we can actually compute the dimension of Bv as a k-vector space! Indeed, we claim that the sequences

i F (p) 0 → H1 −→ Hv+1 −→ Hv → 0. F (p) are exact. That i is the kernel of Hv+1 −→ Hv is clear. Also, the identity FV = [p]G(p) implies that F : G → G(p) is surjective in the sense of fppf abelian sheaves (as this holds for [p] on the (p) F (p) p-divisible group G ), so Hv+1 −→ Hv is surjective in that sheaf sense by pullback. But we know that for finite commutative k-groups, a homomorphism is faithfully flat iff it is surjective in the F (p) sheaf sense. Thus, Hv+1 −→ Hv is faithfully flat and so the above sequence is exact, as claimed. In particular, it follows that

v order(Hv) = (order(H1)) .

But H1 is a ﬁnite group killed by F with an n dimensional lie algebra. We digress to discuss its structure. Lemma. Let H = Spec(R) be a ﬁnite commutative group scheme over k killed by F with n dimensional Lie algebra. Then as a scheme,

p p H = Spec(k[X1, ..., Xn]/(X1 , ..., Xn)).

2 Proof. (Mumford, p. 139) Choose x1, ..., xn ∈ me which generate the cotangent space me/me. Then the map

p p k[X1, ..., Xn]/(X1 , ..., Xn) → R; Xi 7→ xi is surjective. On the other hand, because it is a group scheme, R = Γ(OH ) admits derivations Di so that Di(xj) = 1 mod me (the proof of the existence of “enough invariant differentials” for not necessarily smooth finite flat group schemes is given in [Neron Models, p.100]). Then expanding a1 an monomials X1 ...Xn , 0 ≤ ai < p by the Leibnitz rule, it is straightforward to see that there can be no lower order linear dependences over k (since char(k) = p.) The claim follows.

nv pv pv nv It follows that order(Hv) = p . On the other hand, k[X1, ..., Xn]/(X1 , ..., Xn ) is p - dimensional. Thus, uv is an isomorphism. Passing to the limit, we get

k[[X1, ..., Xn]] → A is an isomorphism, as desired.

12 We interpret the Serre-Tate equivalence in more concrete terms for an abelian variety X/k, where k is a ﬁeld of characteristic p. The formal Lie group corresponding to X(p)0 is none other than the formal group of X/k!

• X/k is smooth. Thus, X,ˆ the completion of X along the identity section is a power series ring over k. Recall that we deﬁned Xb to be the restriction of the functor X to small artinian local k-algebras. Xb[pv] is tautologically isomorphic to

T 7→ X(T )[pv] = X[pv](T ).

But any small artinian point lies of X[pv] automatically lies in X[pv]0 because it is supported at the identity. Thus, Xb[pv] and X[pv]0 are canonically isomorphic. Since any small artinian point is pv-torsion for some v, we get the functor isomorphism

Xb = lim Xb[pv] ∼= lim X[pv]0. → →

Hence, Xˆ corresponds to X(p)0 via the Serre-Tate equivalence.

Life is better with the Serre-Tate equivalence in hand. We deﬁne the dimension of G to be the dimension of the divisible formal Lie group associated to G0.

We digress to discuss a cool result. It’s a nice illustration of how to use the connected etale sequence.

Claim. If (R, m) is a local noetherian ring with residue characteristic p > 0, then the “special ﬁber” functor from p-divisible groups over R to p-divisible groups over the residue ﬁeld is faithful.

The noetherian hypothesis is necessary here. Indeed, for the non-noetherian valuation ring n R = Zp[ζp∞ ], there’s the map Qp/Zp → Gm(p) given by 1/p 7→ ζpn (for a p-power compatible system of choices of ζpn ’s), and this map is an isomorphism on generic fibers but vanishes on the special fiber. Proof. (Brian) Let G and H be p-divisible groups over R, and f : G → H a map whose special fiber f0 : G0 → H0 vanishes. We want to prove that f = 0. We start off knowing that f ⊗R R/m = 0. Suppose that for any artinian quotient A of R that whenever I ⊂ A is an ideal with mI = 0 and that f ⊗R A/I = 0 then f ⊗R A = 0 (∗). If we n assume by induction that f ⊗R R/m = 0, then applying (*) with n+1 n n+1 n n+1 A = R/m ,I = m ⊂ R/m , A/I = R/m shows that f ⊗R R/m = 0. But fv : Gv = Spec(A) → Hv = Spec(B) is the zero map precisely when the corresponding R-Hopf f ∗ algebra map B −→v A factors through B/J, J the augmentation ideal of B. By assumption, ∗ n ∗ J ⊂ ker(fv ) + m B for each n. By Krull’s intersection theorem, this implies that fv factors through B/J, i.e. that fv = 0. Since now we only need to prove the inductive step (*), we can assume for an ideal I in R killed by m that f mod I vanishes. I will prove that f ◦ [p] = 0, so then f = 0 (that this suffices uses

13 that [p] is surjective, which is only true at the level of p-divisible groups, not for ﬁnite ﬂat group schemes).

Functoriality of the connected-etale sequence leads us to ﬁrst consider the etale and connected cases separately, and then we’ll be reduced to proving that there’s no nonzero map from an etale p-divisible group over R to a connected one (which is false without a noetherian condition, by the above example). The case of etale G and H is trivial. Indeed, the equivalence of categories for etale p-divisible ˜ groups says that any map f : G → H corresponds to a π1(R, α)-equivariant map f of free ˜ Zp-modules with a continuous linear actions. But f0 is the same underlying map of Zp-modules (the diﬀerence being the Galois action, given through the isomorphism π1(k, α) → π1(R, α)). So

˜ ˜ f0 = 0 =⇒ f0 = 0 =⇒ f = 0 =⇒ f = 0. For the connected case, we switch viewpoint to that of formal Lie groups, so f corresponds to a map of augmented R-algebras

∗ f : R[[x1, ..., xn]] → R[[y1, ..., yN ]] ∗ such that f (xj) has all coeﬃcients in I (since f mod I = 0) with constant term 0. But then

∗ ∗ ∗ ∗ (f ◦ [p]) (xj) = [p] (f (xj)) = [p] (stuff with I-coefficients with constant term 0), ∗ yet every [p] (xi) has linear terms with coefficient p. So, if we plug into it anything with I-coefficients (and no constant term!) then we get zero since pI = 0 and I2 = 0 (as I is killed by the maximal ideal of R).

j So, the map f : G → H factors uniquely through G −→G Get, the cokernel of G0 −→iG G, say through 0 et 0 et 0 00 0 f : G → H. Since jH ◦ f = f ◦ jG = 0, f factors through iH , the kernel of jH . Say iH ◦ f = f . It clearly suﬃces to prove that f 00 = 0. Thus, we are reduced to prove that f = 0 if G is etale and H is connected, and can assume f mod I = 0.

By replacing the artin local R with a strict henselization, we can assume G is constant, and then that G = Qp/Zp. Thus, f corresponds to a sequence of p-power compatible elements in the groups

ker(Hn(R) → Hn(R/I)). By viewing each coordinate ring OHn as a quotient of the formal power series ring OH , it is clear that the kernel of each map Hn(R) → Hn(R/I) is killed by [p]. Another fundamental quantity associated with finite group schemes H = Spec(B) is the discriminant: disc(H) is defined to be the ideal disc(B) ⊂ R of the finite free R-algebra B.

Lemma. Let 0 → H0 → H → H00 → 0 be an exact sequence of ﬁnite ﬂat R-groups of respective orders m0, m, m00. Then

disc(H) = (disc(H0))m00 (disc(H00))m0 .

Proof. See Rebecca’s notes. Our next immediate goal will be to prove the following theorem:

14 Theorem. Let G = (Gv, iv) be a p-divisible group of height h and dimension n over a complete local noetherian ring R with residue characteristic p. Then

nvphv disc(Gv) = (p ). Proof. The preceeding lemma allows us to simplify our computation of the discriminant in two ways:

• From the deﬁnition of p-divisible groups, we have an exact sequence

i j 0 → G1 −→ Gv+1 −→ Gv → 0.

Thus, it suﬃces to compute the discriminant of G1. • If H is a ﬁnite etale group scheme, then disc(H) = 1. Thus, by the connected-etale sequence, our computation is reduced to the case of connected p-divisible groups.

So, if Γ is the divisible formal Lie group, with coordinate ring A, associated to G, then ∗ G1 = Spec(A/[p] (I)). But by an earlier discussion, if a1, ..., an denotes a basis for A, viewed as a module over itself via 0 another copy A of itself through [p], then a1, ..., an is an R-basis for A/[p](I). Since the discriminant is computed by the determinant of a matrix of traces, it is compatible with reduction. Thus, it will suﬃce to compute discA0/A.

Let Ω, Ω0 be the modules of formal diﬀerentials of A, A0 respectively. They are R-linearly spanned 0 by the dXi, dXi respectively. The map [p]: A0 → A induces d[p]:Ω0 → Ω. A choice of bases for A (resp. A0) determines generators θ (resp. θ0) of ∧nΩ (resp. ∧nΩ0). Thus, ∧ndψ(θ0) = aθ for some a ∈ A.

In the appendix, we show that

discA/A0 = NA/A0 (a).

Grant this fact for now. Choose a basis ωi of “translation invariant differentials”. The existence of such a basis is proven in [Conrad, Leiblich, p. 73], which follows a similar line of reasoning to [Neron Models, p. 100]. For our purposes, it will suffice to note that any invariant differential ω satisfies the property that dµ(ω) = ω ⊕ ω. Since [p] corresponds to multiplication by p in Γ,

0 d[p](ωi) = pωi. Using this particular basis of diﬀerentials to determine θ, it follows that a, from above, equals pn. Thus, from above, it follows that

h n rank 0 A np discA/A0 = NA/A0 (a) = (p ) A = (p ).

15 Duality for p-divisible Groups

As explained earlier, for any p-divisible group G = (Gv, iv) of height h over a commutative ring R, we have an exact sequence

i1,v j1,v 0 → G1 −−→ Gv+1 −−→ Gv → 0. Taking Cartier duals gives the dual exact sequence

j∨ i∨ ∨ 1,v ∨ 1,v ˇ 0 → Gv −−→ Gv+1 −−→ G1 → 0. v ∨ ∨ p ∨ The maps j1,v are the kernels of Gv+1 −→ Gv+1. Also, since duality preserves the order of ﬁnite ∨ ∨ ∨ objects, we get that G = (Gv , j1,v) is a p-divisible group of the same height h. We have the following important theorem relating G and G∨.

Theorem. Let R be a complete, local, noetherian ring with residue ﬁeld k of characteristic p. Let n, n∨ denote the dimensions of G, G∨. Then

n + n∨ = h.

Proof. The dimensions and heights of G, G∨ are the same as the dimensions and heights of ∨ Gk,Gk . Thus, we are reduced to the case where R is a ﬁeld of characteristic p. In the abelian category of fppf abelian group sheaves over k, we have the following commutative diagram

0 −−−→ ker F −−−→ G −−−→F G(p) −−−→ 0     p  y y V y 0 −−−→ 0 −−−→ G −−−→id G −−−→ 0 with exact rows. The snake lemma then yields the exact sequence

0 −−−→ ker F −−−→ ker p −−−→ ker V −−−→ 0. (∗) But we can compute the orders of each term in the exact sequence:

h • ker p = G1 has order p .

• As explained in the proof of the Serre-Tate equivalence, ker F (called H1 in that proof) has order pn.

• By deﬁnition, V is the dual map to F. Thus, because duality is exact, the kernel of V is the dual of the cokernel of

∨ (p) ∨ F : G1 → (G1 ) .

∨ (p) ∨ Since G1 and (G1 ) have the same order, and since orders are multiplicative in exact sequences, the cokernel of F the above map has the same order as its kernel, namely pnˇ.

By multiplicativity or orders applied to (*), it follows that n + n∨ = h, as claimed.

16 Here are some basic instances of this great result.

• Gm(p) has height 1 and dimension 1. It’s dual is Qp/Zp, which has height 1 and dimension 0. • Consideran abelian scheme X of dimension g over R. A couple well known facts over ﬁelds are, remarkably, true over any base whatsoever (Faltings-Chai):

– The dual abelian scheme X0/R always exists. – Duality of abelian schemes is compatible with Cartier duality, i.e.

X(p)∨ = X0(p). Composing this natural isomorphism with the Cartier duality pairing

v 0 v ∼ v v ∨ X[p ] × X [p ] −→ X[p ] × X[p ] → µpv . gives the usual Weil pairing.

Via a generalization of the fact proved earlier about abelian varieties, the formal Lie group corresponding to X(p)0 under the Serre-Tate equivalence is X,b the formal Lie group of X. In particular, it follows that X(p),X0(p) both have height 2g and dimension g. The height of X(p)0 thus assumes values between g to 2g. (It turns out that it can assume all of them.) For example, if X/R is an elliptic curve, then the height is 2 if the elliptic curve has supersingular reduction over the special ﬁber and 1 if it has ordinary reduction.

Appendix

Using the notation from our partial proof of Tate’s discriminant calculation, we want to prove that

discA/A0 = NA/A0 (a). The reference that Tate gives for this fact is inadequate. So during the Wiles Seminars for FLT, Brian devised a proof on his own. I include it here for completeness.

∼ 0 Suppose we have an A-module isomorphism T r : A = HomA0 (A, A ), so T rA/A0 (x) = T r(dx) for some d ∈ A unique up to unit multiple (and independent of choice of T r too). We wish to compute (d). If fi(X1, ..., Xn) = [p](Xi) ∈ R[[X1, ..., Xn]], [p] gives an isomorphism of A0 algebras

∼ 0 A = A [[X1, ..., Xn]]/(Fi), 0 where Fi = fi(X1, ..., Xn) − Xi and ∂Fi/∂Xj = ∂fi/∂Xj.

Claim 2. • (d) = (det(∂fi/∂Xj)).

• discA/A0 = (NA/A0 (det(∂fi/∂Xj)).

17 0 0 0 If θ denotes the A -module generator dX1 ∧ ... ∧ dXn of Ω (resp. θ denotes the A -module 0 0 0 generator dX1 ∧ ... ∧ dXn of Ω ), then

n n 0 n 0 ∧ d[p]: ∧ Ω → ∧ Ω; θ 7→ det(∂fi/∂Xj)θ,

so (a) = (det(∂fi/∂Xj)). By claim 2(i), it follows that

vn pvh vnpvh discA/A0 = (NA/A0 (a)) = ((p ) ) = (p ) where the second last equality folows by choosing a basis of Ω0 consisting of invariant diﬀerentials to compute that a and pvn diﬀer by an A×-multiple.

Claim 2(ii) falls into a more general framework.

Claim 200. Let A be a ring, R an A-algebra which is ﬁnite an free as an R-module. Assume ∼ HomA(R,A) = R as R-mods, and let T rR/A → τ ∈ R under such an isomorphism (so τ is well deﬁned up to R×). Then

discR/A = (NR/A(τ)).

n Proof. Say R = ⊕i=1Aei with πi the ith coordinate projection. By deﬁnition, discR/A is generated ∼ by det(T rR/A(eiej)). Let HomA(R,A) = R take f0 to 1, and choose ri ∈ R such that πi ↔ ri so that πi(x) = f0(rix). Thus,

NR/A(τ) = det(πj(τei))

= det(f0(rjτei))

= det(f0(τ(rjei)))

= det(T rR/A(rjei)). P But {ri} forms an A-basis of R because {πi} forms an A-basis of HomA(R,A). Thus, rj = l ajlel for some (aij) ∈ GLn(A). Then X (T rR/A(rjei)) = ( ajlT rR/A(elei)) = (ajl)(T rR/A(elei)). l

Therefore, NR/A(τ) = unit · det(T rR/A(eiej)), as desired. ∼ 0 Finally, claim 2(i), and our original assumption that A = HomA0 (A, A ), are special cases of the 0 next result, where we ﬁrst observe that {Fi} is a regular sequence in A [[X1, ..., Xn]]. Claim 20. Let O be a complete local noetherian ring,

0 O = O[[T1, ..., Tn]]/(f1, ..., fn) non-zero with {fi} a regular sequence in O[[T1, ..., Tn]]. Then one has an isomorphism

0 ∼ 0 HomO(O , O) = O 0 as O -modules where T rO0/O 7→ det(∂fi/∂Tj).

18 Proof. For this, we use Tate’s results in appendix 3 to Mazzur-Roberts (see references). His 0 theorem A.3 there (with R = O,A = O[[T1, ..., Tn]], fi = fi,C = O ) gives an explicit isomorphism 0 0 HomO(O , O) as O -modules with T rO0/O 7→ β(d) where

0 0 β : O [[T1, ..., Tn]] 7→ O ; Ti 7→ Ti P and fi(T1, ..., Tn) = j bij(Tj − Tj), d = det(bij). P But bij = ∂fi/∂Tj = (Tl − Tl) · stuﬀ, so

β(d) = det(∂fi/∂Tj)(T1, ..., Tn) = det(∂fi/∂Tj).

References

• Bosch, Lutkebohmert, Raynaud. Neron Models. (1990)

• Conrad, B. Diﬀerent Formulations of Gorenstein and Complete Intersection Conditions. Wiles Seminar (1994).

• Conrad, B. Shimura-Taniyama Formula.

• Conrad, B., Leiblich. Galois Representations Arising from p-divisible groups.

• Matsumura. Commutative Ring Theory.

• Mazur, Roberts. Local Euler Characteristics. Inventiones math. 9, 201-234 (1970).

• Mumford. Abelian Varieties.

• Tate. Finite Flat Group Schemes, in “Modular Forms and Fermat’s Last Theorem” (1995).

• Tate. p-divisible Groups.